Amazing! What is the weight of your 308? I'm guessing under 9lbs with the scope.
First, let's get some scope here so you know why people are abusing you. 223 level is basically no felt recoil in and average rifle. 243 Win can be felt. 308 is moderately firm. 300 mag in a hunting weight rifle is definately firm. Heavy recoil starts at like 375 H&H and it would be memorable, but this is also the caliber women and irregular shooters are asked to bring to Africa. 458 Lott, my dream rifle, has recoil. 5 rounds from a bench is likely unbearable, but from shooting positions would likely just be considered "HEAVY." 505 Gibbs, 600 Overkill, 470 NE, 577 NE are all "HEAVY!" I want one so bad!
Personally, I would bet this is 90% of your issue. First, get the rifle solidly on your shoulder and off your arm.
See how the line from shoulder to shoulder is nearly perpendicular. to the bore axis. That is the key. It allows the recoil to travel down the strong muscles of your back. Holding it back into your shoulder with your trigger hand with about 5 lbs of pressure helps also.
A 308 should be comfortable in a 9lb rifle including scope. More weight makes recoil feel like less. In a 15 lb rifle, you will feel little.
Lighter bullets and lower velocities recoil less. Those combined with weight are the primary physical things which affect rifle recoil.
A straighter stock profile putting the action just above the shoulder line will provide the best recoil transfer and reduced recoil. Adding a good recoil pad helps. Many rifles, especially in 308 or lesser calibers come with a rather thin rubber pad. A ~1" pad or larger can be a big help. Also, some rifles have a lot of drop(barrel centerline above shoulder) This can really add to the recoil feeling.
SCOPE POSITION AND EYE RELIEF:
Sounds weird but, I don't trust scopes. If my rifles stock makes me likely to scrunch up around the eyepiece, I hate it. If my scopes eye relief does the same, I basically fear the recoil more that the recoil itself. There is nothing like taking a scope in the ey to make you hate recoil. Scopes with 3.5" or more eye relief are key. Then get the technique and stock adjustments so you are likely to fall into your comfort spot quickly and easily at the bench, prone, etc.
So, take it easy and really play around until you find a way to shoot this thing. I wouldn't mess with extra recoil pads, shooting jackets, etc at this point, but who am I to say!